In the setting described in the Introduction, second order corrections to
the position \(m^\varepsilon (t)\)  of the front are obtained in
[1]}, where it has been shown that the suitably centered and
rescaled front fulfills an invariance principle. Again, the proof takes
advantage of analyzing (REF ) first. Let us note here that in
[2]}, a corresponding invariance principle has been derived for
non-linearities that are either ignition type or bistable; note however,
that – as will be explained below – on a logarithmic in time scale
these fronts behave quite differently from the fronts to (REF )
in our context. For a different and due technical reasons restricted set
of initial conditions, Nolen [3]} has derived a central limit
theorem for the position of the front of the solution to (REF )
by analytic means. The initial condition \(w_0(x,\xi )\)  of [3]} is
required to depend on the randomness of the environment.
